Which Of The Following Represents The Sum Of The Series Below?$S_n=\sum_{k=1}^7[1+(k-1)(2)]A. The Sum Of The First 7 Odd Numbers B. The Sum Of The First 7 Even Numbers C. The Sum Of The 7 Numbers Beginning With 2 D. The Sum Of The 7

Introduction

In mathematics, a series is a sequence of numbers that are added together to find a total sum. The given series, $S_n=\sum_{k=1}^7[1+(k-1)(2)]$, represents a specific sequence of numbers that we need to analyze and find the sum of. In this article, we will break down the series, understand its pattern, and determine which of the given options represents the sum of the series.

Breaking Down the Series

The given series is $S_n=\sum_{k=1}^7[1+(k-1)(2)]$. To understand this series, let's break it down step by step.

• The series starts with $k=1$, which means the first term is $1+(1-1)(2) = 1$.
• The series then moves to $k=2$, which means the second term is $1+(2-1)(2) = 3$.
• This pattern continues until $k=7$, where the seventh term is $1+(7-1)(2) = 13$.

Understanding the Pattern

By analyzing the series, we can see that each term is an odd number. The first term is 1, the second term is 3, the third term is 5, and so on. This pattern continues until the seventh term, which is 13.

Analyzing the Options

Now that we understand the pattern of the series, let's analyze the given options:

• A. The sum of the first 7 odd numbers: This option represents the sum of the first 7 odd numbers, which are 1, 3, 5, 7, 9, 11, and 13.
• B. The sum of the first 7 even numbers: This option represents the sum of the first 7 even numbers, which are 2, 4, 6, 8, 10, 12, and 14.
• C. The sum of the 7 numbers beginning with 2: This option represents the sum of the first 7 numbers that begin with 2, which are 2, 4, 6, 8, 10, 12, and 14.
• D. The sum of the 7 numbers: This option is too vague and does not provide any specific information about the numbers.

Conclusion

Based on our analysis, we can conclude that the sum of the series $S_n=\sum_{k=1}^7[1+(k-1)(2)]$ represents the sum of the first 7 odd numbers. This is because each term in the series is an odd number, and the sum of the first 7 odd numbers is 1 + 3 + 5 + 7 + 9 + 11 + 13 = 49.

The Final Answer

Therefore, the correct answer is:

• A. The sum of the first 7 odd numbers

Additional Information

To further understand the series, let's calculate the sum of the first 7 odd numbers:

1 + 3 + 5 + 7 + 9 + 11 + 13 = 49

Q: What is the series $S_n=\sum_{k=1}^7[1+(k-1)(2)]$?

A: The series $S_n=\sum_{k=1}^7[1+(k-1)(2)]$ represents a sequence of numbers that are added together to find a total sum. The series starts with $k=1$ and continues until $k=7$, where each term is calculated using the formula $1+(k-1)(2)$.

Q: What is the pattern of the series?

A: The pattern of the series is that each term is an odd number. The first term is 1, the second term is 3, the third term is 5, and so on. This pattern continues until the seventh term, which is 13.

Q: What is the sum of the series?

A: The sum of the series is the total value of all the terms added together. Based on our analysis, the sum of the series is 1 + 3 + 5 + 7 + 9 + 11 + 13 = 49.

Q: Which of the given options represents the sum of the series?

A: Based on our analysis, the correct answer is:

• A. The sum of the first 7 odd numbers

Q: Why is option A the correct answer?

A: Option A is the correct answer because each term in the series is an odd number, and the sum of the first 7 odd numbers is 49.

Q: What is the difference between the series and the sum of the first 7 odd numbers?

A: The series $S_n=\sum_{k=1}^7[1+(k-1)(2)]$ represents a specific sequence of numbers, while the sum of the first 7 odd numbers is a general concept that represents the total value of all the odd numbers added together.

Q: How can I calculate the sum of the series?

A: To calculate the sum of the series, you can use the formula for the sum of an arithmetic series, which is:

Sum = (n/2)(a + l)

where n is the number of terms, a is the first term, and l is the last term.

In this case, n = 7, a = 1, and l = 13. Plugging these values into the formula, we get:

Sum = (7/2)(1 + 13) = 49

Q: What is the significance of the series?

A: The series $S_n=\sum_{k=1}^7[1+(k-1)(2)]$ is a simple example of a mathematical concept that can be used to model real-world problems. Understanding the series and its pattern can help you to solve problems and make predictions in various fields, such as finance, engineering, and science.

Q: Can I use the series to model real-world problems?

A: Yes, you can use the series to model real-world problems. For example, you can use the series to model the growth of a population, the cost of a project, or the revenue of a business.

Q: How can I apply the series to real-world problems?

A: To apply the series to real-world problems, you need to identify the pattern of the series and use it to model the problem. You can then use the formula for the sum of an arithmetic series to calculate the total value of the series.

Conclusion

In conclusion, the series $S_n=\sum_{k=1}^7[1+(k-1)(2)]$ represents a sequence of numbers that are added together to find a total sum. The series has a specific pattern, and the sum of the series is 49. Understanding the series and its pattern can help you to solve problems and make predictions in various fields.